The paper describes a construction of abstract polytopes from Cayley graphs of symmetric groups. Given any connected graph G with p vertices and q edges, we associate with G a Cayley graph G(G) of the symmetric group S p and then construct a vertex-transitive simple polytope of rank q, the graphicahedron, whose 1-skeleton (edge graph) is G(G). The graphicahedron of a graph G is a generalization of the well-known permutahedron; the latter is obtained when the graph is a path. We also discuss symmetry properties of the graphicahedron and determine its structure when G is small.

A carousel is a dynamical system that describes the movement of an equilateral linkage in which the midpoint of each rod travels parallel to it. They are closely related to the floating body problem. We prove, using the work of Auerbach, that any figure that floats in equilibrium in every position is drawn by a carousel. Of special interest are such figures with rational perimetral density of the floating chords, which are then drawn by carousels. In particular, we prove that for some perimetral densities the only such figure is the circle, as the problem suggests.

No abstract

In this paper we study the topology of transversals to a family of convex sets as a subset of a Grassmanian manifold. This topology seems to be ruled by a combinatorial structure which we call a separoid. With these combinatorial objects and the topological notion of virtual transversal we prove a Borsuk-Ulam-type theorem which has as a corollary a generalization of Hadwiger's theorem.

An r-segment hypergraph H is a hypergraph whose edges consist of r consecutive integer points on line segments in R 2 . In this paper, we bound the chromatic number χ(H) and covering number τ (H) of hypergraphs in this family, uncovering several interesting geometric properties in the process. We conjecture that for r ≥ 3, the covering number τ (H) is at most (r − 1)ν(H), where ν(H) denotes the matching number of H. We prove our conjecture in the case where ν(H) = 1, and provide improved (in fact, optimal) bounds on τ (H) for r ≤ 5. We also provide sharp bounds on the chromatic number χ(H) in terms of r, and use them to prove two fractional versions of our conjecture. 1 arXiv:1807.04826v1 [math.CO]